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Chapter 2: Problem 28

Solve and check. Label any contradictions or identities. $$\frac{2}{3} t-1=5$$

Short Answer

Expert verified
The solution is \( t = 9 \). Substituting back into the original equation verifies it.

Step by step solution

01

Isolate the variable term

First, add 1 to both sides of the equation to isolate the term involving the variable on one side. \[ \frac{2}{3}t - 1 + 1 = 5 + 1 \]This simplifies to: \[ \frac{2}{3}t = 6 \]
02

Solve for the variable

Next, to solve for the variable \(t\), multiply both sides of the equation by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \): \[ t = 6 \times \frac{3}{2} \]This simplifies to: \[ t = 9 \]
03

Check the solution

Substitute \( t = 9 \) back into the original equation to verify the solution:\[ \frac{2}{3} \times 9 - 1 = 5 \]First, calculate \( \frac{2}{3} \times 9 \):\[ 6 - 1 = 5 \]This simplifies to:\[ 5 = 5 \]Since both sides of the equation are equal, the solution \( t = 9 \) is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolate the Variable
Solving a linear equation often begins with isolating the variable. This means we want to get the variable on one side of the equation and everything else on the other. In our exercise, we start with \(\frac{2}{3} t - 1 = 5\). To isolate the term involving \(t\), we need to remove the constant \(-1\) from the left side.

We do this by adding \(1\) to both sides: \(\frac{2}{3} t - 1 + 1 = 5 + 1\). The negative and positive ones cancel out, leaving us with: \(\frac{2}{3} t = 6\).

Now we have an expression where the variable term is isolated. This is our first and crucial step toward solving the equation.
Reciprocal
The next key concept is using the reciprocal. When a variable is being multiplied by a fraction, we use the reciprocal of that fraction to solve for the variable.

In our equation, we have \(\frac{2}{3} t = 6\). To isolate \(t\), we multiply both sides by the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\):

\[t = 6 \times \frac{3}{2}\]
When we perform the multiplication on the right side, we get:

\[t = 9\]
This step is essential because multiplying by the reciprocal effectively cancels out the fraction on the left and leaves us with the variable \(t\) alone.
Substitution Method
After solving for the variable, it鈥檚 important to verify the solution using the substitution method. This means substituting the solution back into the original equation to check its correctness.

In our case, we substitute \( t = 9 \) back into the original equation \( \frac{2}{3} t - 1 = 5 \).

First, we calculate \( \frac{2}{3} \times 9 \):
\[\frac{2}{3} \times 9 = 6\]
Then we subtract 1:
\[6 - 1 = 5\]
Since the left side of the equation equals the right side (\(5 = 5\)), the solution \(t = 9\) is verified.

This step is crucial to ensure that our solution is correct and satisfies the original equation.
Verify Solution
Verifying the solution is an important final step. This gives us confidence that our answer is accurate.

In our example, we found that \( t = 9 \). By substituting it back into the original equation \( \frac{2}{3} t - 1 = 5 \) and performing the calculations, we confirmed that both sides match.

To summarize the verification steps:
  • Substitute the solution \( t = 9 \) back into the original equation.
  • Simplify the left-hand side: \( \frac{2}{3} \times 9 - 1 = 6 - 1 = 5 \).
  • Compare both sides of the equation: \( 5 = 5 \).
This consistent result tells us that our initial solution is indeed correct and that the equation has been solved accurately.

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